Question

Although there is legal protection for "whistle blowers" - employees who report illegal or unethical activities in the workplace- it has been reported that approximately \(23 \%\) of those who reported fraud suffered reprisals such as demotion or poor performance ratings. Suppose the probability that an employee will fail to report a case of fraud is .69. Find the probability that an employee who observes a case of fraud will report it and will subsequently suffer some form of reprisal.

Short answer

Answer: Approximately 7.13%.

Step-by-step solution

Find the probability of reporting a fraud (Event A)

To find the probability of reporting a fraud (Event A), we can use the complement rule since we know the probability of not reporting a fraud (A') is 0.69. P(A) = 1 - P(A') P(A) = 1 - 0.69

Calculate the joint probability of reporting a fraud and suffering a reprisal (Event A and B)

Now we have the probability of reporting a fraud (P(A)), and we also have the conditional probability of suffering a reprisal given that the employee reported the fraud (P(B|A)). We can use the formula for joint probability to find the probability we're looking for. P(A and B) = P(B|A) * P(A) Plug in the probabilities from Step 1. P(A and B) = 0.23 * (1 - 0.69)

Calculate the final probability

Now, let's compute the joint probability to find the probability that an employee who observes a case of fraud will report it and will subsequently suffer some form of reprisal. P(A and B) = 0.23 * (1 - 0.69) P(A and B) ≈ 0.23 * 0.31 P(A and B) ≈ 0.0713 So, the probability that an employee who observes a case of fraud will report it and will subsequently suffer some form of reprisal is approximately 7.13%.

Key concepts

Joint Probability

Joint probability refers to the probability of two events occurring together. In this situation, we want to know the probability that an employee will both report a fraud and experience reprisal. These two events, reporting the fraud and the occurrence of reprisal, are dependent on each other.

To calculate this, we use the formula for joint probability:

• \( P(A \text{ and } B) = P(B|A) \cdot P(A) \)

This equation involves the conditional probability of event B, given that event A has occurred, and the probability of event A.

By determining the likelihood of one event and considering its effect on another, joint probability gives us a comprehensive view of how likely it is for multiple events to happen simultaneously. In our exercise, the joint probability results in a 7.13% chance.

Complement Rule

The Complement Rule is a basic principle in probability that helps in finding the probability of an event not happening by using the probability of the event happening. It asserts that the probability of an event's complement (the event not occurring) plus the probability of the event itself equals one.

Mathematically, it is expressed as:

• \( P(A') = 1 - P(A) \)

where \( P(A') \) is the probability of the event not occurring.

In our case, knowing that the probability of an employee not reporting the fraud is 0.69, we can use the Complement Rule to find the probability of an employee actually reporting the fraud. By simply subtracting 0.69 from 1, you find that the probability of reporting is 0.31, a 31% likelihood.

Conditional Probability

Conditional Probability examines the probability of an event occurring given that another event has already happened. This concept is key in scenarios where outcomes are contingent on previous occurrences. It is expressed mathematically as:

• \( P(B|A) = \frac{P(A \text{ and } B)}{P(A)} \)

where \( P(B|A) \) is the probability of event B given event A has occurred.

In the context of our problem, it is noted that approximately 23% of employees who report fraud experience some form of reprisal. Thus, the conditional probability \( P(B|A) \) is 0.23, as it represents the likelihood of facing repercussions if fraud is reported.

Understanding conditional probability is crucial, especially when dealing with real-world situations where one event influences the likelihood of another, helping to better comprehend interconnected probabilities.